AA65.pdf 6.5 Completing the Square 1. Converting from vertex form to standard form involves expanding the square of the binomial, distributing a, and then isolating y. What method does converting from standard form to vertex form use? Completing the Square y k = a(x h) 2 to y = ax 2 + bx + c (Vertex form) (Standard form) 2. What is a perfect square trinomial? A trinomial that is the square of a binomial 3. Draw a picture of (x + h) 2 = x 2 + 2hx + h 2 4. Study example 1 5. Now try the following problem in your notes: a. What number should be added to x 2 + 5x to make a perfect square trinomial? Draw a picture to represent x 2 + 5x. Square half the coefficient of x.?? x 2.5? 2.5? x Dec 9 9:28 AM 1
6. Theorem about completing the square. 7. Study example 2 8. Now try the following problem in your notes: a. Rewrite the equation below in vertex form: i. y = x 2 + 18x + 90 y k = a(x h) 2 1. Rewrite with all x terms alone on one side of the equation y = x 2 + 18x + 90 90 90 2. If a = 1 add to both sides of the equation y 90 = x 2 + 18x +81 + 81 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be y 9 = x 2 + 18x + 81 y 9 = (x + 9) 2 Using the calculator we can find the vertex. The value of a will be the same as the value of a in the original equation. k = 9, h = 9, a = 1 Dec 9 9:32 AM 2
8. Now try the following problem in your notes: a. Rewrite the equation below in vertex form: y = x 2 11x + 4 ii. y = x 2 11x + 4 y k = a(x h) 2 Rewrite with the x terms alone on one side of the equation y = x 2 11x + 4 4 4 x 5.5 5.5 x If a = 1 add or y 4 = x 2 11x to both sides of the equation The trinomial should then be able to be rewritten as a binomial squared. The first term should be x and the second term should be y = x 2 11x y = (x ) 2 y + 26.25 = (x 5.5) 2 k = 26.25, h = 5.5, a = 1 Using the calculator we can find the vertex. The value of a will be the same as the value of a in the original equation. Dec 9 2:03 PM 3
9. Study example 3 10. Now try the following problem in your notes: a. Rewrite the equation below in vertex form: i. y = 3x 2 12x + 1 y k = a(x h) 2 Rewrite with the x terms alone on one side of the equation y = 3x 2 12x + 1 1 1 y 1 = 3x 2 12x if a does not equal 1, divide both sides by a y 1 = 3x 2 12x 3 3 Now add to both sides of the equation The trinomial should then be able to be rewritten as a binomial squared. The first term of the binomial should be x and the second term of teh binomial should be In order to get the equation in vertex form multiply both sides by the original a to clear fractions y k = a(x h) 2 k = 11, h = 2, a = 3 Using the calculator we can find the vertex. The value of a will be the same as the value of a in the original equation. Dec 9 9:35 AM 4
9. Study example 3 10. Now try the following problem in your notes: a. Rewrite the equation below in vertex form: ii. y = 5x 2 + 4x 3 y k = a(x h) 2 Rewrite with the x terms alone on one side of the equation y = 5x 2 + 4x 3 +3 +3 y + 3 = 5x 2 + 4x Divide both sides by a y + 3 = 5x 2 + 4x 5 5 Now add to both sides of the equation In order to get the equation in vertex form multiply both sides by a The trinomial should then be able to be rewritten as a binomial squared. The first term should be x and the second term should be Using the calculator we can find the vertex and the value of a will be the same as the value of a in the original equation. y +2.2 = 5(x.4) 2 k = 2.2, h =.4, a = 5 Dec 9 9:35 AM 5
11. Now try the following problem in your notes: a. Suppose a ball is thrown straight up from a height of 4 feet with an initial velocity of 50 feet per second. What is the maximum height of the ball? h(t) = 16t 2 + 50t + 4 The maximum height of 43.0625 feet is reached 1.5625 seconds after the ball was thrown. Dec 9 9:35 AM 6
Carefully sketch the graph of y = 4x 2 6x + 1.75 Identify all important properties of this quadratic function: Mar 14 7:39 AM 7
Carefully sketch the graph of y + 3 = 2(x.5) 2 Identify all important properties of this quadratic function: Mar 14 7:39 AM 8
Carefully sketch the graph of y + 3 = 2(x.5) 2 Identify all important properties of this quadratic function: Carefully sketch the graph of y + 3 = 2(x.5) 2 Identify all important properties of this quadratic function: Mar 14 7:39 AM 9
y = 4x 2 6x + 1.75 Mar 14 8:21 AM 10
Rewrite each of the following quadratic functions in the other form, sketch each one, and identify all significant properties studied: y 1 = 4(x + 2) 2 y = 2x 2 4x 2.25 Mar 16 7:21 AM 11
y = 2x 2 4x 2.25 Mar 16 1:23 PM 12
Determine a value for a and b to make the statement true. x 2 10x + 25 = ( x + a) 2 a = 4x 2 +12x + 9 = ( 2x + a) 2 a = x 2 + bx + 81 = ( x + a) 2 a = b = x 2 +bx + 100 = ( x + a) 2 a = b = x 2 8x + b = ( x + a) 2 a = b = x 2 + 12x + b = ( x + a) 2 a = b = Dec 18 8:03 AM 13
Rewrite each of the following quadratic functions in the other form, sketch each one, and identify all significant properties studied: y = 3x 2 +24x 36 y = 3x 2 +24x 36 Mar 17 7:29 AM 14
Notes 6 5 Review: Perfect Square Trinomial is the result of the square of a binomial (x + 11) 2 = x 2 + 22x + 121 y = x 2 + 6x + 10 y = x 2 + 6x +? Completing the Square Algebraically To Go From Standard To Vertex Form a = 1 1. Subtract/add the constant to both sides of the equation. 2. Divide the coefficient of the x term by 2 and square it. Then add that number to both sides of the equation. 3. Simplify both sides. (right side of equation should be a binomial squared) 4. Vertex : Jul 30 6:11 PM 15
Try: y = x 2 12x + 24 1.Subtract/add the constant to both sides of the equation. 2. Divide the coefficient of the x term by 2 and square it. Then add that number to both sides of the equation. 3. Simplify both sides. (right side of equation should be a binomial squared) 4. Vertex : Jul 30 6:21 PM 16
y = 2x 2 20x + 57 Completing the Square Algebraically To Go From Standard To Vertex Form a 1 1. Subtract/add the constant to both sides of the equation. 2. Multiply both sides by the reciprocal of the coefficient of x 2. (don t simplify left side of the equation ) 3. Divide the coefficient of the x term by 2 and square it. Then add that number to both sides of the equation. 4. Simplify the right side of the equation into a binomial squared. Vertex: 5. Multiply both sides of the equation by the reciprocal of the number originally multiplied on the left side. 6. Simplify the left side of the equation. Do not distribute on the right side. Jul 30 6:25 PM 17
Try: y = 6x 2 18x 5 Jul 30 6:28 PM 18
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1. Rewrite with all x terms alone on one side of the equation 2. If a = 1 add to both sides of the equation If a does not equal 1, divide both sides by a now add to both sides of the equation 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be Dec 9 3:57 PM 22
1. Rewrite with all x terms alone on one side of the equation 2. If a = 1 add to both sides of the equation If a does not equal 1, divide both sides by a now add to both sides of the equation 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be Dec 9 3:57 PM 23
1. Rewrite with all x terms alone on one side of the equation 2. If a = 1 add to both sides of the equation If a does not equal 1, divide both sides by a now add to both sides of the equation 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be Dec 9 3:57 PM 24
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1. Rewrite with all x terms alone on one side of the equation 2. If a = 1 add to both sides of the equation If a does not equal 1, divide both sides by a now add to both sides of the equation 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be Dec 9 3:57 PM 26
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1. Rewrite with all x terms alone on one side of the equation 2. If a = 1 add to both sides of the equation If a does not equal 1, divide both sides by a now add to both sides of the equation 3. The trinomial should now be a perfect square trinomial. It can be rewritten as a binomial squared. The first term should be x and the second term should be Dec 8 9:28 AM 45
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Attachments AA65.pdf Chapter 6 WS 6 5.pdf